examday

/TMUA

Consider a set of distinct integers $S$.

An element $N$ in $S$ is $S$-good if and only if there exist 2 other elements in $S$ that have a mean of $N$.

An element $N$ in $S$ is $S$-unique if and only if there exist exactly 2 other elements in $S$ that have a mean of $N$.

Let $G,U$ denote the numbers of $S$-good, $S$-unique elements respectively.

Which of the following must be true?

• I. For all $n\ge3$, there exists a set $S$ of size $n$ where $U=n-2$.

• II. If $U=G$, then there exist two elements $x,y$ in $S$ where

  $\displaystyle |x-y|\ge\frac{U(U+1)}2.$

• III. For all integers $g\ge0$, there exists a set $S$ where $U=0$ and $G=g$.